(c) If \(E\)  and \(F\)  are real locally convex spaces,
then we call \(f\)  real analytic, resp., \(C^\omega \) ,
if for each point \(x \in U\)  there exists an open neighborhood
\(V \subseteq E_ and a holomorphic map \) f V F with
\(f__{U \cap V} = f\vert _{U \cap V}\)  (cf. [1]}).
The advantage of this definition, which differs from the one in
[2]}, is that it works nicely for non-complete spaces,
any analytic map is smooth,
and the corresponding chain rule holds without any condition
on the underlying spaces, which is the key to the definition of
analytic manifolds (see [3]} for details).

